In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.

Let ${\mathcal{T}}_{n}$ denote the set of log canonical thresholds of pairs $(X,Y)$, with $X$ a nonsingular variety of dimension $n$, and $Y$ a nonempty closed subscheme of $X$. Using non-standard methods, we show that every limit of a decreasing sequence in ${\mathcal{T}}_{n}$ lies in ${\mathcal{T}}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${\mathcal{T}}_{n}$ is closed in $\mathbf{R}$; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check...

We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles ${\mathcal{M}}_{L}$ that arise as the...

The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. The functional behavior of these invariants is related to the set-theoretic behavior of base loci.

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